A thin piece of wire 40 metres long is cut into two pieces. One piece is used to form a circle with radius r , and the other is used to form square. No wire if left over. Which of the following represents the total area, in square meters, of the circular and the square regions in terms of r?

a. Pi.r^2 (Formula for an area of a circle)
b. Pi.r^2 + 10
c. Pi.r^2 + 1/4(Pi^2.r^2)
d. Pi.r^2 + (40 - 2.Pi.r)^2
e. Pi.r^2 + (10 - 0.5.Pi.r)^2

The way I did this question was to denote each piece 20m long. This made a square with sides of 5m and a circle with a radius of 20m. So, the total area I got was Pi.400 + 25 (metres squared). The correct answer was e., and I guessed e, but that answer does not equal my total area.

Thanks!
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The answer shld be E

2.pi.r+4s(side of the sqare) = 40
s=40-2.pi.r)/4==> (10-.5pi.r)^2

Therefore
Total area = Area of the circle (pi.r^2)+(10-.5pi.)^2 ==> E

Ah, I see now where I made my mistake. The 20 I used is not the radius, but rather the circumference of the circle. So r = 10/Pi.

Now, my total area = Pi.r^2 + 25, and answer e. works!

Thanks a lot! :)

How would you complete this problem without picking numbers? Or, is picking numbers the way to go for this problem? If so, what flags should I have seen to alert me that this is a picking number type problem.

This is how to do without the using actual numbers.

Area of circle = PI.r^2
side of the square = 40-2.PI.r/4 = 10 - 0.5.PI.r
Area of square = (10 - 0.5.PI.r)^2 =

Add both the areas above. The answer works out to E.

so you have a total length of 40.

you use part of that length to make a circle with radius r. this circle will have circumference 2πr, so the leftover amount with which to make the square is 40 - 2πr.

therefore, each side of the square is one-fourth of (40 - 2πr), or 10 - 0.5πr.
the area of the square is thus (10 - 0.5πr)^2, so the answer (e) follows.

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by the way, i would not recommend number picking on this problem, as the formulas are terribly awkward - not to mention the fact that you actually don't have to do that much work to derive the formula. the hard part is getting past the conceptual barrier of realizing that you have to subtract the circle's circumference from 40 in order to find out the amount of wire left to make the square.